Shape Keys are inactive in Edit Mode
Shape keys work by storing two states for your mesh and allowing you to blend between the two. First off though I would caution just adding shape keys with no armature. They can have uses without armatures, but they are most commonly used to either correct bad/weird deformation or to add additional details to the deformation that are hard to make with an armature.

Here's a quick rundown of how shape keys work. This isn't extensive by any means, but should get you up and running with shape keys.

First step is to start in object mode, then add a shape key with the little '+' icon. This will create a shape key called Basis. This is the first state that blender stores, and as the name suggests, it is the default, original shape of your mesh, as if it had no shape keys.

Next, add another shape key. You can name this one whatever you want by clicking on the name. This will be the second state that is saved for your model. By default, the slider will be set to 0, which is fine.

Now, with the second key selected, go into edit mode. You can edit different keys by selecting whichever key you want to edit. Note that you do have to be in object mode to select a different key, which is why they are greyed out in your case. Now you can edit your mesh to be whatever state you want to blend between. You could, for example, close an eye, or create some wrinkles that would appear when the mouth of your model would open, or whatever you like. Once you are happy with your model, go back to object mode. You will notice that it will go back to your original model.

The slider under your shape key will now do something, and if you adjust to anything other than 0, you will see your model blend between the two states. You can continue to edit and add as many shape keys as you like as many times as you like.

One thing to note, in addition to what I cautioned at the start, is how shape keys deform your mesh. Unlike an armature, which connects the vertices to a bone and makes them follow the bone, shape key deformation is linear. Basically what that means is each vertex will go in a straight line between its position in the Basis key and the additional key. To give an example, if you made a shape key to bend a characters elbow, and you went into edit mode and rotated the mesh about the elbow, so that in edit mode the motion is proper, the shape key would not pay any attention to how you moved the vertices, just the start and end positions.

Here is a link to the Blender Manual page for shape keys (link) which you can go through to find out more about each setting and the workflows behind it, but I would suggest trying things by yourself a bit first to see what you can figure out.